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A087116 Number of maximal groups of consecutive zeros in binary representation of n. 15

%I #38 Oct 18 2021 13:12:02

%S 1,0,1,0,1,1,1,0,1,1,2,1,1,1,1,0,1,1,2,1,2,2,2,1,1,1,2,1,1,1,1,0,1,1,

%T 2,1,2,2,2,1,2,2,3,2,2,2,2,1,1,1,2,1,2,2,2,1,1,1,2,1,1,1,1,0,1,1,2,1,

%U 2,2,2,1,2,2,3,2,2,2,2,1,2,2,3,2,3,3,3,2,2,2,3,2,2,2,2,1,1,1,2,1,2,2

%N Number of maximal groups of consecutive zeros in binary representation of n.

%C The following four statements are equivalent: a(n) = 0; n = 2^k - 1 for some k; A087117(n) = 0; A023416(n) = 0.

%H Reinhard Zumkeller, <a href="/A087116/b087116.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F a(n) = A033264(n) for n > 0 since strings of 0's alternate with strings of 1's. - _Jonathan Sondow_, Jan 17 2016

%F a(n) = a(2*n + 1) = a(4*n + 2) - 1, if n > 0. - _Michael Somos_, Nov 04 2016

%F a(n) = A069010(A003817(n)-n) for n > 0. - _Chai Wah Wu_, Nov 04 2016

%e G.f. = 1 + x^2 + x^4 + x^5 + x^6 + x^8 + x^9 + 2*x^10 + x^11 + x^12 + x^13 + x^14 + ...

%t a[n_] := SequenceCount[IntegerDigits[n, 2], {Longest[0..]}];

%t Table[a[n], {n, 0, 101}] (* _Jean-François Alcover_, Oct 18 2021 *)

%o (Haskell)

%o a087116 0 = 1

%o a087116 n = f 0 n where

%o f y 0 = y

%o f y x = if r == 0 then g x' else f y x'

%o where (x', r) = divMod x 2

%o g z = if r == 0 then g z' else f (y + 1) z'

%o where (z', r) = divMod z 2

%o -- _Reinhard Zumkeller_, Mar 31 2015

%o (PARI)

%o a(n) = if (n == 0, 1, hammingweight(bitxor(n, n>>1)) >> 1);

%o vector(102, i, a(i-1)) \\ _Gheorghe Coserea_, Sep 17 2015

%o (Python)

%o def A087116(n):

%o return sum(1 for d in bin(n)[2:].split('1') if len(d)) # _Chai Wah Wu_, Nov 04 2016

%Y Cf. A087118, A087119, A087120, A023416, A007088.

%Y Cf. A023416, A087117.

%Y Cf. A033264, A003817, A069010.

%Y Essentially the same as A033264.

%K nonn

%O 0,11

%A _Reinhard Zumkeller_, Aug 14 2003

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Last modified March 28 16:34 EDT 2024. Contains 371254 sequences. (Running on oeis4.)